Real-rooted atlas

A245962 → D+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(5 + x - 4\,x^{2} - 7\,n - 5\,n x + 4\,n x^{2} + n^{2} + 4\,n^{2} x + n^{3}\right)\,P_{n}(x) = \left(11 - x - 18\,x^{2} - 8\,x^{3} - 18\,n - 15\,n x + 10\,n x^{2} + 8\,n x^{3} + 4\,n^{2} + 14\,n^{2} x + 8\,n^{2} x^{2} + 3\,n^{3} + 2\,n^{3} x\right)\,P_{n-1}(x) + \left(-10 - 53\,x - 91\,x^{2} - 64\,x^{3} - 16\,x^{4} - 20\,n - 56\,n x - 52\,n x^{2} - 16\,n x^{3} - 5\,n^{2} - 9\,n^{2} x - 4\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
111
2527
37310
41512229
52625556
64860212131
785126567274
8151264152322601
926553136910291276
1046310508653104522735
1180420351936847165115798
12139038884206219255560212293
13239373328892538216772471325936
144105136781839612712473991077254619
Technical details
Node id
n-3ce21439d055e53f · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-635fb24c49f1a904

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half affine: slope 1, intercept 0)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:   1;
  1:   1;
  2:   5,   2;
  3:   7,   3;
  4:  15,  12,   2;
  5:  26,  25,   5;
  6:  48,  60,  21,   2;
  7:  85, 126,  56,   7;
  8: 151, 264, 152,  32, 2;
  9: 265, 531, 369, 102, 9;
%F Formula
The row polynomials satisfy (n^3+n^2-7*n+5+(4*n^2-5*n+1)*x+(4*n-4)*x^2)*P(n,x) = (3*n^3+4*n^2-18*n+11+(2*n^3+14*n^2-15*n-1)*x+(8*n^2+10*n-18)*x^2+(8*n-8)*x^3)*P(n-1,x) - (5*n^2+20*n+10+(9*n^2+56*n+53)*x+(4*n^2+52*n+91)*x^2+(16*n+64)*x^3+16*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((3*n^3 + 4*n^2 - 18*n + 11 + (2*n^3 + 14*n^2 - 15*n - 1)*x + (8*n^2 + 10*n - 18)*x^2 + (8*n - 8)*x^3)*P[n - 1, x] - (5*n^2 + 20*n + 10 + (9*n^2 + 56*n + 53)*x + (4*n^2 + 52*n + 91)*x^2 + (16*n + 64)*x^3 + 16*x^4)*D[P[n - 1, x], x])/(n^3 + n^2 - 7*n + 5 + (4*n^2 - 5*n + 1)*x + (4*n - 4)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((3*n^3 + 4*n^2 - 18*n + 11 + (2*n^3 + 14*n^2 - 15*n - 1)*x + (8*n^2 + 10*n - 18)*x^2 + (8*n - 8)*x^3)*v[i - 1] - (5*n^2 + 20*n + 10 + (9*n^2 + 56*n + 53)*x + (4*n^2 + 52*n + 91)*x^2 + (16*n + 64)*x^3 + 16*x^4)*deriv(v[i - 1], x))/(n^3 + n^2 - 7*n + 5 + (4*n^2 - 5*n + 1)*x + (4*n - 4)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

References

  1. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.